AC-Close: Efficiently Mining Approximate Closed Itemsets by Core Pattern Recovery

Transcription

1 : Efficiently Mining Approximate Closed Itemsets by Core Pattern Recovery Hong Cheng Philip S. Yu Jiawei Han University of Illinois at Urbana-Champaign IBM T. J. Watson Research Center {hcheng3, Abstract Recent studies have proposed methods to discover approximate frequent itemsets in the presence of random noise. By relaxing the rigid requirement of exact frequent pattern mining, some interesting patterns, which would previously be fragmented by exact pattern mining methods due to the random noise or measurement error, are successfully recovered. Unfortunately, a large number of uninteresting candidates are explored as well during the mining process, as a result of the relaxed pattern mining methodology. This severely slows down the mining process. Even worse, it is hard for an end user to distinguish the recovered interesting patterns from these uninteresting ones. In this paper, we propose an efficient algorithm to recover the approximate closed itemsets from core patterns. By focusing on the so-called core patterns, integrated with a top-down mining and several effective pruning strategies, the algorithm narrows down the search space to those potentially interesting ones. Experimental results show that substantially outperforms the previously proposed method in terms of efficiency, while delivers a similar set of interesting recovered patterns. 1. Introduction Despite the exciting progress in frequent itemset mining [1, 4, 12, 8, 2], an intrinsic problem with the exact mining methods is the rigid definition of support. In real applications, a database contains random noise or measurement error. For example, in a customer transaction database, random noise could be caused by an out-of-stock item, promotions or special events. Measurement error could be caused by noise from experiments, uncertainty involved in discretization. Such random noise can distort the true underlying patterns. In the presence of noise, the exact frequent itemset mining algorithms will discover multiple fragmented patterns, but miss the longer true patterns. Recent studies [11, 5] try to recover the true patterns in the presence of noise, by allowing a small fraction of errors in the itemsets. While some interesting patterns are recovered, a large number of uninteresting candidates are explored during the mining process. Such candidates are uninteresting in the sense that they correspond to no true frequent patterns in the unobserved true transaction database. They are generated as a result of the relaxed mining approach. Let s first examine Example 1 and the definition of the approximate frequent itemset (AFI) [5]. Example 1 Table 1 shows a transaction database D. Let the minimum support threshold min sup = 3; ɛ r = 1/3, where ɛ r is the fraction of noise (0s) allowed in a row (i.e., in a supporting transaction); and ɛ c = 1/2, where ɛ c is the fraction of noise allowed in a column (i.e., an item in the set of supporting transactions). TID Burger Coke Diet Coke Table 1. A Sample Purchase Database D According to [5], {burger, coke, diet coke} is an AFI with support 4, although its exact support is 0 in D. It is deemed uninteresting since coke and diet coke are seldom purchased together in reality. Those 0s in D are not caused by random noise, but reflect the real data distribution. 1

2 To tackle such a problem, we propose to recover the approximate frequent itemsets from core patterns. Intuitively, an itemset x is a core pattern if its exact support in the noisy database D is no less than αs, where α [0, 1] is a core pattern factor, and s is min sup. Based on the problem formulation, we design an efficient algorithm to mine the approximate closed itemsets. A top-down mining strategy is exploited, which makes full use of the pruning power of min sup and closeness and thus, narrows down the search space dramatically. Experimental studies show that, substantially outperforms AFI in terms of efficiency, while delivers a similar set of the recovered true patterns. The remainder of the paper is organized as follows. In Section 2, preliminary concepts are introduced. In Section 3, the approximate itemset mining and pruning techniques are introduced. The algorithm is presented in Section 4. Experimental results are reported in Section 5 and related work is discussed in Section 6. We conclude our study in Section 7. 2 Preliminary Concepts Let a transaction database D take the form of an n m binary matrix. Let I = {i 1, i 2,..., i m } be a set of all items. A subset of I is called an itemset. Let T be the set of transactions in D. Each row of D is a transaction t T and each column is an item i I. A transaction t supports an itemset x, if for each item i x, the corresponding entry D(t, i) = 1. An itemset x is frequent if the fraction of transactions supporting it is no less than a user-specified threshold min sup. We use different notations to distinguish the support concept of exact frequent itemsets from that of approximate frequent itemsets. The exact support of an itemset x is denoted as sup e (x). The exact supporting transactions of x is denoted as T e (x). For an approximate itemset x, the support is denoted as sup a (x) and its supporting transaction set is denoted as T a (x). Definition 1 An itemset x is a core pattern in the noisy database D if the exact support of x, sup e (x) αs, where α [0, 1] is the core pattern factor and s is the min sup threshold. In the presence of random noise, the true frequent patterns would still show some clues, though their supports are affected by the noise. To reflect such a property, we use the core pattern mechanism to recover the true patterns. In our definition of approximate frequent itemset, we adopt from [5] the two variables ɛ r and ɛ c as the error thresholds. The definition is given as follows. Definition 2 Let ɛ r, ɛ c [0, 1] and the transaction database be D. An approximate frequent itemset x is (1) a core pattern with sup e (x) αs, and (2) if there exists a set of transactions T a (x) T with T a (x) s T, s is the min sup threshold, and the following two conditions hold: 1. i T a (x), 2. j x, 1 x 1 T a (x) D(i, j) (1 ɛ r ) j x i T a (x) D(i, j) (1 ɛ c ) ɛ r and ɛ c are referred to as row error threshold and column error threshold, respectively. To further reduce the number of approximate frequent itemsets discovered, we define the concept of approximate closed itemsets. Definition 3 An approximate frequent itemset x is closed if there exists no itemset y such that (1) y is a proper superset of x, and (2) sup a (y) sup a (x). For an approximate pattern x and its superset y, it is possible that sup a (y) sup a (x), as caused by the 0-extension effect [5]. In this case, x is nonclosed since y subsumes x in both directions of items and transactions. The problem of approximate closed itemset mining from core patterns is the mining of all itemsets which are (1) core patterns w.r.t. α; (2) approximate frequent itemsets w.r.t. ɛ r, ɛ c and min sup; and (3) closed. 3 Approximate Closed Itemset Mining In this section, we study the problem of mining approximate closed itemsets with ɛ r, ɛ c and min sup, and develop several pruning strategies as well. 3.1 Candidate Approximate Itemset Generation To discover the approximate itemsets, we first mine the set of core patterns with min sup = αs. These patterns are treated as the initial seeds for possible further extension to approximate frequent itemsets. Let C be the set of core patterns. A lattice L is built over C. Example 2 is used to illustrate the process. Example 2 Table 2 shows a transaction database D. Let ɛ r = 0.5, ɛ c = 0.5, min sup = 3, and α = 1/3. Mining the core patterns from D with the absolute support α min sup = 1 generates 15 core patterns. Figure 1 shows the lattice of core patterns. 2

3 TID a b c d Table 2. A Transaction Database D 3.2 Pruning by ɛ c After we identify the candidate approximate itemsets which satisfy min sup and ɛ r, the next step is to check each candidate x w.r.t. the column error threshold ɛ c. According to Definition 2, it needs to scan the approximate transaction set T a (x). However, a pruning strategy, referred to as ɛ c early pruning, allows us to identify some candidates which violate the ɛ c constraint without scanning T a (x). This pruning strategy is stated as follows. Lemma 2 Let x = i 1...i n be an itemset and the exact support of a single item i k x in D be sup e (i k ), k [1, n]. Let the support of the approximate pattern x be sup a (x). If Figure 1. The Lattice L of Core Patterns For core pattern {a, b, c, d}, the number of 0s allowed in a supporting transaction is = 2. Traverse upward in the lattice for 2 levels (i.e., levels 2 and 3), which constitute the extension space for {a, b, c, d}. The extension space is defined as follows. Definition 4 For a core pattern y with size l, the extension space, denoted as ES(y), is a set of subpatterns of y from level ( l (1 ɛ r ) ) to level (l 1) in the lattice, given ɛ r [0, 1]. Because of the ɛ r fraction of errors allowed in a transaction, for a core itemset y and each sub-pattern x ES(y), any transaction supporting x also approximately supports y. According to this property, the transaction set T a (y) is the union of the transaction set T e (x), x ES(y). Thus, for {a, b, c, d}, its transaction set T a ({a, b, c, d}) is the union of the transaction sets of all itemsets at levels 2 and 3 of Figure 1. This property is stated formally in Lemma 1. Lemma 1 For a candidate approximate itemset y and each pattern x ES(y), the approximate transaction set T a (y) is computed by taking the union of the exact transaction set T e (x). To summarize, the steps to identify candidate approximate itemsets include: (1) For each core pattern y, identify its extension space ES(y) according to ɛ r ; (2) For each x ES(y), take the union of T e (x); and (3) Check against min sup. Keep those which satisfy min sup as candidate approximate frequent itemsets. i k x, sup e(i k ) sup a (x) < (1 ɛ c) satisfies, then x cannot pass the ɛ c check. The proof is omitted due to the space limit. The ɛ c early pruning is effective especially when ɛ c is small, or there exists an item in x with very low exact support in D. If the pruning condition is not satisfied, a scan on T a (x) is performed for the ɛ c check. 3.3 Top-down Mining and Pruning by Closeness In this section, we show that an efficient top-down search strategy can be exploited as the mining framework, thus enabling effective pruning by the closeness definition and the min sup threshold. A top-down mining starts with the largest pattern in L and proceeds level by level, in the size decreasing order of core patterns. Let s look at an example. Example 3 Mining on the lattice L starts with the largest pattern {a, b, c, d}. Since the number of 0s allowed in a transaction is 2, its extension space includes patterns at levels 2 and 3. The transaction set T a ({a, b, c, d}) is computed by the union operation. Since there exists no approximate itemset which subsumes {a, b, c, d}, it is an approximate closed itemset. When the mining proceeds to level 3, for example, the size-3 pattern {a, b, c}. The number of 0s allowed in a transaction is = 1, so its extension space includes its sub-patterns at level 2. Since ES({a, b, c}) ES({a, b, c, d}) holds, T a ({a, b, c}) T a ({a, b, c, d}) holds too. In this case, after the computation on {a, b, c, d}, we can prune {a, b, c} without actual computation with either of the following two conclusions: (1) if {a, b, c, d} satisfies the 3

4 min sup threshold and the ɛ c constraint, then no matter whether it is closed or non-closed, {a, b, c} can be pruned because it will be a non-closed approximate itemset; or (2) if {a, b, c, d} does not satisfy the min sup, then {a, b, c} can be pruned because it will not satisfy the min sup threshold either. In the first condition, we say no matter whether {a, b, c, d} is closed or non-closed, {a, b, c} is non-closed. This is because, if {a, b, c, d} is closed, then {a, b, c} is subsumed by {a, b, c, d} and thus is non-closed; if {a, b, c, d} is non-closed, then there must exist a closed approximate super-pattern which subsumes {a, b, c, d}, and then subsumes {a, b, c} as well. Thus {a, b, c} is nonclosed. We refer to the pruning technique in Example 3 as forward pruning, which is formally stated in Lemma 3. Lemma 3 If (k + 1) ɛ r = k ɛ r + 1, after the computation on a size-(k + 1) pattern is done, all its size-k sub-patterns in the lattice L can be pruned with either of the following two conclusions: (1) if the size- (k +1) pattern satisfies min sup and ɛ c, then the size-k patterns can be pruned because they are non-closed; or (2) if the size-(k + 1) pattern does not satisfy min sup, then the size-k patterns can be pruned because they do not satisfy min sup either. Forward pruning, naturally integrated with the topdown mining, can reduce the search space dramatically due to the min sup and the closeness constraint. Another pruning strategy, called backward pruning, is proposed as well to ensure the closeness constraint, as stated in Lemma 4. Lemma 4 If a candidate approximate pattern x satisfies min sup, ɛ r and ɛ c, it has to be checked against each approximate closed itemset y where x y. If there exists no approximate closed itemset y such that x y and sup a (x) sup a (y), then x is an approximate closed itemset. 4 Algorithm Integrating the top-down mining and the various pruning strategies, we proposed an efficient algorithm to mine the approximate closed itemsets from the core patterns, presented in Algorithm 1. 5 Experimental Study A systematic study was performed to evaluate the algorithm. We compared it with AFI 1. In the 1 Since the AFI implementation has not been released yet, we implemented this algorithm to the best of our understanding. Input: D, min sup = s, ɛ r, ɛ c, α Output: ACI: approximate closed itemsets 1: C = genfreqitemset(d, αs); 2: L = buildlattice(c); 3: k = max level of L; 4: C k = {x x L and size(x) = k}; 5: repeat 6: for x C k 7: ES = genextensionspace(x, L); 8: T a (x) = uniontransaction(es); 9: if (x not satisfies s or ɛ c ) 10: C k = C k {x}; 11: if( k ɛ r = (k 1) ɛ r + 1) 12: C k 1 = forwardprune(c k 1, x); 13: end for 14: L k = backwardprune(c k, ACI); 15: ACI = ACI L k ; 16: k = k 1; 17: until k = 0 18: return ACI Algorithm 1: The Algorithm first group of experiments, the efficiency of AFI and were tested. In the second group, the result quality of AFI and were tested on synthetic datasets with a controlled fraction of random noise embedded and with known underlying patterns. 5.1 Scalability The IBM synthetic data generator is used to generate datasets for test. A dataset T10.I100.D20K [1] is generated with 20K transactions, 100 distinct items and an average of 10 items per transaction. Figure 2 (a) shows the running time of both algorithms by varying min sup with ɛ r = ɛ c = α = 0.8 is used in. As shown in the figure, runs much faster than AFI. Figure 2 (b) shows the running time of both algorithms by varying ɛ r and ɛ c. To reduce the parameter space, we set ɛ = ɛ r = ɛ c. min sup = 0.8% and α = 0.8 are used. The running time of AFI increases very quickly as ɛ increases while the efficiency of is not affected much by ɛ. Figure 2 (c) shows the running time of by varying the core pattern factor α. ɛ r = ɛ c = 0.20 is used. As α decreases, the core pattern set becomes larger. Therefore, more candidates are generated for approximate itemset mining and the computation time increases. Nevertheless, is shown to be very 4

5 AFI 10 3 AFI 16 min_sup=0.8% min_sup=1.0% min_sup=2.0% min_sup (%) ε=ε r =ε c α (a) Varying min sup (b) Varying ɛ r, ɛ c (c) Varying α Figure 2. Running Time of AFI and, on T10.I100.D20K 10 3 AFI AFI AFI Number of Transactions (K) Number of Items Avg Items Per Transaction (a) Varying Number of Transactions (b) Varying Number of Items (c) Varying Avg Number of Items Per Transaction Figure 3. Running Time of AFI and, Vary Database Statistics efficient even when α is set to as low as 0.3. Figure 3 shows the scalability tests by varying different statistics of the transaction database. min sup = 1%, ɛ r = ɛ c = 0.20 and α = 0.8 are used. Figure 3 (a) shows the running time by varying the number of transactions. The number of distinct items is 100 and the average items per transaction is 10. Since both algorithms only perform the union or intersection operations on the transaction id lists without scanning the database repeatedly, increasing the transaction number does not affect the performance much. Figure 3 (b) shows the running time by varying the number of distinct items in the database. The number of transactions is 10K and the average transaction length is 10. When the number of items is large, the database becomes sparse. So both algorithms run more efficiently. As the number of items decreases, the number of approximate frequent itemsets increases. Figure 3 (b) shows that the running time of AFI increases more rapidly than that of. Figure 3 (c) shows the running time by varying the average number of items per transaction. The number of transactions is 10K and the number of distinct items is 100. As the average transaction length increases, the combination between items increases exponentially. The figure shows that the running time of AFI increases much faster. 5.2 Quality We test and compare the quality of mining results by AFI and. A dataset D t, T10.I100.D20K, is used as the noise-free dataset. Noise is introduced to flip each entry of D t with a probability p. The noisy dataset is denoted as D. Exact frequent pattern mining is applied to D t and the results are treated as the ground truth, denoted as F true. In addition, AFI and are applied to D and the approximate itemsets are treated as the recovered patterns, denoted as F apr. Two evaluation metrics precision and recall are used as the measure, as defined below. precision = F true F apr, recall = F true F apr F apr F true Table 3 shows the quality comparison between AFI and with p = ɛ r = ɛ c = 0.20 and α = 0.8 are used. As we can see, AFI achieves the same 5

6 recall values as but AFI also generates some false positive patterns which do not appear in F true. In contrast, the precision of is 100%. Table 3. Precision and Recall of Mining Result by AFI and, Noise Level p = 0.05 min sup(%) Precision Recall AFI AFI Table 4. Precision and Recall of the Mining Result by, varying α, p = 0.05 α min sup=0.8% min sup=1.0% Precision Recall Precision Recall AFI Table 4 shows the result quality of at different α values, at a noise level p = ɛ r = ɛ c = 0.20 is used. At the bottom row of the table, the precision and recall of AFI is shown as the comparison baseline. As shown in Table 4, a large α value achieves the same recall as a small α when the noise level is low. 6 Related Work Error-tolerant itemset (ETI) mining was first studied by [11]. Two error-tolerant models, weak ETI and strong ETI are proposed. Some heuristics are used to greedily search for ETIs. Liu et al. [5] proposed an approximate frequent itemset (AFI) model which controls errors of two directions in the data matrices formed by transactions and items. Other related studies on approximate frequent patterns include [6, 3, 7, 10, 9]. [6] showed that approximate association rules are interesting and useful. [3] proposed the concept of free-sets and led to an errorbound approximation of frequencies. The goal of [7] is to derive a compact representation condensed frequent pattern base. In the support envelope study [10], a symmetric ETI model was proposed. [9] proposed to mine the dense itemsets the itemsets with a sufficiently large submatrix that exceeds a given density threshold. 7 Conclusions In this paper, we propose an effective algorithm for discovering approximate closed itemsets from transaction databases in the presence of random noise. Core pattern is proposed as a mechanism for efficiently identifying the potentially true patterns. Experimental study demonstrates that our method can efficiently discover a large fraction of true patterns while avoiding generating false positive patterns. References [1] R. Agrawal and R. Srikant. Fast algorithms for mining association rules. In Proc. VLDB 94, pages [2] R. J. Bayardo. Efficiently mining long patterns from databases. In Proc. SIGMOD 98, pages [3] J. Boulicaut, A. Bykowski, and C. Rigotti. Approximation of frequency queries by means of free-sets. In Proc. PKDD 00, pages [4] J. Han, J. Pei, and Y. Yin. Mining frequent patterns without candidate generation. In Proc. SIGMOD 00, pages [5] J. Liu, S. Paulsen, X. Sun, W. Wang, A. Nobel, and J. Prins. Mining approximate frequent itemsets in the presence of noise: Algorithm and analysis. In Proc. SDM 06, pages [6] H. Mannila and H. Toivonen. Multiple uses of frequent sets and condensed representations. In Proc. KDD 96, pages [7] J. Pei, G. Dong, W. Zou, and J. Han. Mining condensed frequent pattern bases. Knowledge and Information Systems, 6(5): , [8] J. Pei, J. Han, and R. Mao. CLOSET: An efficient algorithm for mining frequent closed itemsets. In Proc. DMKD 00, pages [9] J. Seppänen and H. Mannila. Dense itemsets. In Proc. KDD 04, pages [10] M. Steinbach, P. Tan, and V. Kumar. Support envelopes: A technique for exploring the structure of association patterns. In Proc. KDD 04, pages [11] C. Yang, U. Fayyad, and P. S. Bradley. Efficient discovery of error-tolerant frequent itemsets in high dimensions. In Proc. KDD 01, pages [12] M. J. Zaki. Scalable algorithms for association mining. IEEE Trans. Knowledge and Data Engineering, 12(2): ,